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October 1st, 2019, 12:37 PM   #1
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Question Can the length of this triangle be solved using congruence?

I found this riddle in my book. Supposedly it should be solved without requiring trig.

The problem is as follows:
A CTV broadcasting tower is resting atop a square flat base in the highest hill of Taipei. The tower is supported by five cables which are held tight to the ground by four retention screws labeled $\textrm{A}$, $\textrm{C}$, $\textrm{D}$, $\textrm{E}$, $\textrm{F}$ as indicated in the picture. The radio technician notices that the length between $\textrm{HC= 3 feet}$, meanwhile $3AB=5BC$ and $m\angle ABH= 3m \angle HBC$. From this information find the distance between $H$ and $A$.


The alternatives given on my book are:

$\begin{array}{ll}
1.&\textrm{13 feet}\\
2.&\textrm{11 feet}\\
3.&\textrm{9 feet}\\
4.&\textrm{10 feet}\\
\end{array}$

What I attempted to do is shown in the figure from below.



But other than just putting what is stated in the problem I could not reach any further in my observations. I tried hard to imagine different ways to arrange the triangles in a manner that they could be used as congruence (from which I noticed one). But other than that I couldn't find more.

Therefore can somebody help me to find the answer using mainly simple Euclidean geometry, i.e. constructions or tying it to congruence or similarity? But please do not just go any further because this problem is supposedly able to be solved without requiring advanced math tools or trigonometry.

I will really appreciate it if somebody could give me a hand with this problem.

Last edited by skipjack; October 1st, 2019 at 02:02 PM.
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October 1st, 2019, 01:56 PM   #2
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Your diagram show a dashed line that bisects angle ABC.
The angle bisector theorem implies 3(AH - 3 feet) = 5(3 + 3) feet, so AH = 13 feet.
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October 7th, 2019, 03:02 AM   #3
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Thumbs up It was in front of me all the time, thanks

Quote:
Originally Posted by skipjack View Post
Your diagram show a dashed line that bisects angle ABC.
The angle bisector theorem implies 3(AH - 3 feet) = 5(3 + 3) feet, so AH = 13 feet.
Yay! I can't believe the answer was in front of me all the time. I missed the fact that you mentioned the angle bisector, after reading that explanation made it all simple. Thanks!
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